# Derivation of torque on general current distribution

• user1139
In summary, -The current densities ( curl and divergence) are zero. -The two expressions are not equal in general. -The first expression can be simplified using integration by parts.
user1139
How do I simplify the expression

$$\int_{\mathcal{V'}}\int_{\mathcal{V}}\frac{\mathbf{\mathrm{x}}\times[\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}[\mathbf{\mathrm{J_2(x)}}\cdot(\mathbf{\mathrm{x}-\mathbf{\mathrm{x'}}})]]}{|\mathbf{\mathrm{x}}-\mathbf{\mathrm{x'}}|^3}\,\mathrm{d^3}x\,\mathrm{d^3}x'$$

to

$$\int_{\mathcal{V'}}\int_{\mathcal{V}}\frac{\mathbf{\mathrm{J_2(\mathbf{\mathrm{x}})}}\times\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}}{|\mathbf{\mathrm{x}}-\mathbf{\mathrm{x'}}|}\,\mathrm{d^3}x\,\mathrm{d^3}x'$$

I was stuck after rewriting the first expression as

$$-\int_{\mathcal{V'}}\int_{\mathcal{V}}[\mathbf{\mathrm{x}}\times\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}] \left[\mathbf{\mathrm{J_2(x)}\cdot\nabla\left(\frac{1}{|\mathbf{\mathrm{x}-\mathbf{\mathrm{x'}}}|}\right)} \right] \,\mathrm{d^3}x\,\mathrm{d^3}x'$$

Delta2
What do we know about the current densities ##\mathbf{J_1},\mathbf{J_2}##? I guess that their divergence is zero that is $$\nabla\cdot\mathbf{J_1}=\nabla\cdot\mathbf{J_2}=0$$. Anything else do we know? Do we know anything about $$\nabla\times\mathbf{J_1},\nabla\times\mathbf{J_2}$$
My thinking tells me that the two expressions aren't equal in the general case. Perhaps you have some typos? Maybe you forgot to put a ##\nabla## operator somewhere in the second expression?

Both current densities are that of steady current. There was no information on the curls of the current densities. I checked again and the second expression does not have any error.

Delta2
You have ##\vec{\nabla} \cdot \vec{J}_1=\vec{\nabla} \cdot \vec{J}_2=0##. Then you have
$$\vec{\nabla} \frac{1}{|\vec{x}-\vec{x}'|}=-\vec{\nabla}' \frac{1}{|\vec{x}-\vec{x}'|}.$$
Then you can use integration by parts wrt. to the integral over ##\vec{x}'##...

Last edited:
I am not sure of the first step. Would you mind writing out the integration by parts?

Delta2

## 1. What is torque and how is it related to current distribution?

Torque is a measure of the rotational force applied to an object. In the context of current distribution, torque is the force that causes an object to rotate due to the interaction of a magnetic field with the current flowing through the object.

## 2. What is the equation for torque on a general current distribution?

The equation for torque on a general current distribution is given by T = I x B, where T is the torque, I is the current, and B is the magnetic field.

## 3. How is the direction of torque determined for a current distribution?

The direction of torque is determined by the right-hand rule, where the thumb points in the direction of the current and the fingers curl in the direction of the magnetic field. The direction of the torque is perpendicular to both the current and the magnetic field.

## 4. Can the torque on a current distribution be increased?

Yes, the torque on a current distribution can be increased by increasing the current or the strength of the magnetic field. Additionally, changing the orientation of the current or the magnetic field can also affect the torque.

## 5. What are some real-world applications of the derivation of torque on general current distribution?

The derivation of torque on general current distribution has various applications in fields such as electrical engineering, physics, and mechanical engineering. It is used in the design of electric motors, generators, and other devices that rely on the interaction of magnetic fields and current. It is also important in understanding the behavior of electric circuits and in the study of electromagnetism.

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